In Exercises 61–64, find the domain of each function.f(x) = √(x/(...

Question

In Exercises 61–64, find the domain of each function.f(x) = √(x/(2x - 1) - 1)

Accepted Answer

Hey everyone here we are asked to find the domain of the function which is the square root of X divided by the quantity of 10 X minus five minus three. And now here we want to start by finding where X is undefined. So we'll look at this denominator here of the fraction and note that when we divide by zero, the result is undefined. So we can take this denominator of 10 X minus five and set it equal to zero to find where X is undefined. So now solving for X, we can just add five to both sides, giving us 10, X is equal to five. And now to get X alone will divide both sides by 10, which gives us X is equal to one half. And now we know that one half cannot be included within the domain of the function because that is where it is undefined. And so next we can take a look at this entire expression underneath the square root and note that it must be positive because if it was negative then we're taking a square root of the negative which is an undefined value. So here we know that this expression X divide by 10, X minus five minus three must be greater than or equal to zero. And now using these two pieces of information, we can write the domain, we have found so far using interval notation, we have X is within one half to positive infinity and now infinity is always closed by a parenthesis because it's an open interval and one half is also going to be closed by a parenthesis because this is where X is undefined, so it cannot be equal to one half. And now from here we can start solving our inequality to find more information about our domain. So solving for X, we can first add three to both sides, giving us x divided by the quantity of 10, X -5 is greater than or equal to three. And now we can multiply both sides by 10, x minus five to get rid of the denominator. So we have X is greater than or equal to 30 X minus 15. And now to get X on one side will subtract 30 X from both sides of the expression. So we have negative 29 X is greater than or equal to negative 15. And now to get X alone will divide both sides by negative 29. But since we're a negative, we have to flip the inequality symbol here. So we have X is less than or equal to 15, 20 nights. And now we can write this inequality using interval notation and we will get X is within negative infinity because X is less than 2. 15, 29th. Now, first, let's note that the infinity sign is again closed by a parentheses, but this 15, 29th is closed by a bracket because X can be equal to this value. And now from here we can combine the two intervals we found and we will get X is within negative infinity to 29th and one half to infinity. And now we just want to know really quickly that this you here is a union symbol just combining these two intervals. And next we want to see that this 15 29th is greater than this one half here. So this 15 29th will be the upper bound of the function, so we can simplify this interval and get X is really within one half as our minimum value to 15 29th as our maxim Value. And again noting that 1529th is closed by a bracket because X can be equal to this value but one half is undefined, so it is closed by a parentheses. Therefore our answer here is choice d with a domain of one half to 1520 nights. Thanks for watching. I hope you found this video helpful and I'll see you again next time.

In Exercises 61–64, find the domain of each function.f(x) = √(x/(2x - 1) - 1)

Key Concepts

Domain of a Function

Square Root Function

Rational Expressions

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